libm/math/lgamma_r.rs
1/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 *
12 */
13/* lgamma_r(x, signgamp)
14 * Reentrant version of the logarithm of the Gamma function
15 * with user provide pointer for the sign of Gamma(x).
16 *
17 * Method:
18 * 1. Argument Reduction for 0 < x <= 8
19 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
20 * reduce x to a number in [1.5,2.5] by
21 * lgamma(1+s) = log(s) + lgamma(s)
22 * for example,
23 * lgamma(7.3) = log(6.3) + lgamma(6.3)
24 * = log(6.3*5.3) + lgamma(5.3)
25 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
26 * 2. Polynomial approximation of lgamma around its
27 * minimun ymin=1.461632144968362245 to maintain monotonicity.
28 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
29 * Let z = x-ymin;
30 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
31 * where
32 * poly(z) is a 14 degree polynomial.
33 * 2. Rational approximation in the primary interval [2,3]
34 * We use the following approximation:
35 * s = x-2.0;
36 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
37 * with accuracy
38 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
39 * Our algorithms are based on the following observation
40 *
41 * zeta(2)-1 2 zeta(3)-1 3
42 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
43 * 2 3
44 *
45 * where Euler = 0.5771... is the Euler constant, which is very
46 * close to 0.5.
47 *
48 * 3. For x>=8, we have
49 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
50 * (better formula:
51 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
52 * Let z = 1/x, then we approximation
53 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
54 * by
55 * 3 5 11
56 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
57 * where
58 * |w - f(z)| < 2**-58.74
59 *
60 * 4. For negative x, since (G is gamma function)
61 * -x*G(-x)*G(x) = PI/sin(PI*x),
62 * we have
63 * G(x) = PI/(sin(PI*x)*(-x)*G(-x))
64 * since G(-x) is positive, sign(G(x)) = sign(sin(PI*x)) for x<0
65 * Hence, for x<0, signgam = sign(sin(PI*x)) and
66 * lgamma(x) = log(|Gamma(x)|)
67 * = log(PI/(|x*sin(PI*x)|)) - lgamma(-x);
68 * Note: one should avoid compute PI*(-x) directly in the
69 * computation of sin(PI*(-x)).
70 *
71 * 5. Special Cases
72 * lgamma(2+s) ~ s*(1-Euler) for tiny s
73 * lgamma(1) = lgamma(2) = 0
74 * lgamma(x) ~ -log(|x|) for tiny x
75 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
76 * lgamma(inf) = inf
77 * lgamma(-inf) = inf (bug for bug compatible with C99!?)
78 *
79 */
80
81use super::{floor, k_cos, k_sin, log};
82
83const PI: f64 = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */
84const A0: f64 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */
85const A1: f64 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */
86const A2: f64 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */
87const A3: f64 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */
88const A4: f64 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */
89const A5: f64 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */
90const A6: f64 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */
91const A7: f64 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */
92const A8: f64 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */
93const A9: f64 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */
94const A10: f64 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */
95const A11: f64 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */
96const TC: f64 = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */
97const TF: f64 = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */
98/* tt = -(tail of TF) */
99const TT: f64 = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */
100const T0: f64 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */
101const T1: f64 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */
102const T2: f64 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */
103const T3: f64 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */
104const T4: f64 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */
105const T5: f64 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */
106const T6: f64 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */
107const T7: f64 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */
108const T8: f64 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */
109const T9: f64 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */
110const T10: f64 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */
111const T11: f64 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */
112const T12: f64 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */
113const T13: f64 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */
114const T14: f64 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */
115const U0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
116const U1: f64 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */
117const U2: f64 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */
118const U3: f64 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */
119const U4: f64 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */
120const U5: f64 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */
121const V1: f64 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */
122const V2: f64 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */
123const V3: f64 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */
124const V4: f64 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */
125const V5: f64 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */
126const S0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
127const S1: f64 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */
128const S2: f64 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */
129const S3: f64 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */
130const S4: f64 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */
131const S5: f64 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */
132const S6: f64 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */
133const R1: f64 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */
134const R2: f64 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */
135const R3: f64 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */
136const R4: f64 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */
137const R5: f64 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */
138const R6: f64 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */
139const W0: f64 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */
140const W1: f64 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */
141const W2: f64 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */
142const W3: f64 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */
143const W4: f64 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */
144const W5: f64 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */
145const W6: f64 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
146
147/* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */
148fn sin_pi(mut x: f64) -> f64 {
149 let mut n: i32;
150
151 /* spurious inexact if odd int */
152 x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */
153
154 n = (x * 4.0) as i32;
155 n = div!(n + 1, 2);
156 x -= (n as f64) * 0.5;
157 x *= PI;
158
159 match n {
160 1 => k_cos(x, 0.0),
161 2 => k_sin(-x, 0.0, 0),
162 3 => -k_cos(x, 0.0),
163 0 | _ => k_sin(x, 0.0, 0),
164 }
165}
166
167#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
168pub fn lgamma_r(mut x: f64) -> (f64, i32) {
169 let u: u64 = x.to_bits();
170 let mut t: f64;
171 let y: f64;
172 let mut z: f64;
173 let nadj: f64;
174 let p: f64;
175 let p1: f64;
176 let p2: f64;
177 let p3: f64;
178 let q: f64;
179 let mut r: f64;
180 let w: f64;
181 let ix: u32;
182 let sign: bool;
183 let i: i32;
184 let mut signgam: i32;
185
186 /* purge off +-inf, NaN, +-0, tiny and negative arguments */
187 signgam = 1;
188 sign = (u >> 63) != 0;
189 ix = ((u >> 32) as u32) & 0x7fffffff;
190 if ix >= 0x7ff00000 {
191 return (x * x, signgam);
192 }
193 if ix < (0x3ff - 70) << 20 {
194 /* |x|<2**-70, return -log(|x|) */
195 if sign {
196 x = -x;
197 signgam = -1;
198 }
199 return (-log(x), signgam);
200 }
201 if sign {
202 x = -x;
203 t = sin_pi(x);
204 if t == 0.0 {
205 /* -integer */
206 return (1.0 / (x - x), signgam);
207 }
208 if t > 0.0 {
209 signgam = -1;
210 } else {
211 t = -t;
212 }
213 nadj = log(PI / (t * x));
214 } else {
215 nadj = 0.0;
216 }
217
218 /* purge off 1 and 2 */
219 if (ix == 0x3ff00000 || ix == 0x40000000) && (u & 0xffffffff) == 0 {
220 r = 0.0;
221 }
222 /* for x < 2.0 */
223 else if ix < 0x40000000 {
224 if ix <= 0x3feccccc {
225 /* lgamma(x) = lgamma(x+1)-log(x) */
226 r = -log(x);
227 if ix >= 0x3FE76944 {
228 y = 1.0 - x;
229 i = 0;
230 } else if ix >= 0x3FCDA661 {
231 y = x - (TC - 1.0);
232 i = 1;
233 } else {
234 y = x;
235 i = 2;
236 }
237 } else {
238 r = 0.0;
239 if ix >= 0x3FFBB4C3 {
240 /* [1.7316,2] */
241 y = 2.0 - x;
242 i = 0;
243 } else if ix >= 0x3FF3B4C4 {
244 /* [1.23,1.73] */
245 y = x - TC;
246 i = 1;
247 } else {
248 y = x - 1.0;
249 i = 2;
250 }
251 }
252 match i {
253 0 => {
254 z = y * y;
255 p1 = A0 + z * (A2 + z * (A4 + z * (A6 + z * (A8 + z * A10))));
256 p2 = z * (A1 + z * (A3 + z * (A5 + z * (A7 + z * (A9 + z * A11)))));
257 p = y * p1 + p2;
258 r += p - 0.5 * y;
259 }
260 1 => {
261 z = y * y;
262 w = z * y;
263 p1 = T0 + w * (T3 + w * (T6 + w * (T9 + w * T12))); /* parallel comp */
264 p2 = T1 + w * (T4 + w * (T7 + w * (T10 + w * T13)));
265 p3 = T2 + w * (T5 + w * (T8 + w * (T11 + w * T14)));
266 p = z * p1 - (TT - w * (p2 + y * p3));
267 r += TF + p;
268 }
269 2 => {
270 p1 = y * (U0 + y * (U1 + y * (U2 + y * (U3 + y * (U4 + y * U5)))));
271 p2 = 1.0 + y * (V1 + y * (V2 + y * (V3 + y * (V4 + y * V5))));
272 r += -0.5 * y + p1 / p2;
273 }
274 #[cfg(debug_assertions)]
275 _ => unreachable!(),
276 #[cfg(not(debug_assertions))]
277 _ => {}
278 }
279 } else if ix < 0x40200000 {
280 /* x < 8.0 */
281 i = x as i32;
282 y = x - (i as f64);
283 p = y * (S0 + y * (S1 + y * (S2 + y * (S3 + y * (S4 + y * (S5 + y * S6))))));
284 q = 1.0 + y * (R1 + y * (R2 + y * (R3 + y * (R4 + y * (R5 + y * R6)))));
285 r = 0.5 * y + p / q;
286 z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */
287 // TODO: In C, this was implemented using switch jumps with fallthrough.
288 // Does this implementation have performance problems?
289 if i >= 7 {
290 z *= y + 6.0;
291 }
292 if i >= 6 {
293 z *= y + 5.0;
294 }
295 if i >= 5 {
296 z *= y + 4.0;
297 }
298 if i >= 4 {
299 z *= y + 3.0;
300 }
301 if i >= 3 {
302 z *= y + 2.0;
303 r += log(z);
304 }
305 } else if ix < 0x43900000 {
306 /* 8.0 <= x < 2**58 */
307 t = log(x);
308 z = 1.0 / x;
309 y = z * z;
310 w = W0 + z * (W1 + y * (W2 + y * (W3 + y * (W4 + y * (W5 + y * W6)))));
311 r = (x - 0.5) * (t - 1.0) + w;
312 } else {
313 /* 2**58 <= x <= inf */
314 r = x * (log(x) - 1.0);
315 }
316 if sign {
317 r = nadj - r;
318 }
319 return (r, signgam);
320}